I just popped my extra AX7 in there and the dwell/zero buzz is gone!
So, exactly what was my issue now? This afternoon I was trying a few things out. I would clip a jumper on to one side of the 1K5 resistor and jump the other side to the other side of the resistor and that would quiet it too. I guess that was just like attaching pin-8 to ground... the path of least resistance and all that. At my dads suggestion I clipped an additional 1K5 resistor in parallel to the 1k5 and that also solved it (depending on where I held the jumped resistor).

So, I guess it is a screwy tube? But I should not run the amp with the 12AX7 in place of a 12AT7 full time, right? That will be a problem? And why exactly is that if that is the case.

Well, I used that 12AT7 for a while with no trouble so it seems like I must have fried it or broken it at some point. I'll order a new AT7 anyway. Glad that I can use the AX7 without worrying about it. I'm happy that this issue is finally resolved.

Okay. Interesting development. I bought a brand new ECC81 (12AT7) tube and put it in... same DWELL buzz. So it was not a bad tube... but something in the circuit that reacts badly to a 12AT7 but works well with at 12AX7.

1) loose socket pins on that socket - in which case (with the amp off and unplugged and the filter caps drained) try wiggling the socket pins with your fingers to see if any of them are loose with the tube plugged in. Sometimes you can push loose pins further on to be 'more tight', or you could re-tension them with a small screwdriver; or

Couldn't detect any dead pins on the socket. Very strange. Whatever differences there are INSIDE a 12AT7, it's just not compatible with my circuit? Weird. I'm not going to buy another tube just to answer this question. Maybe a cheapo chinese at7 or something used from ebay.

The main electrical differences between a 12AT7 and a 12AX7 are the lower dynamic plate resistance and higher transconductance and lower amplification factor of the former (which translates into higher plate-current, lower input-sensitivity/more clean headroom, and lower gain for a 12AT7), and a typically lower h-k voltage tolerance in the AT, all other things being equal. With a plate resistor of ~100k and cathode resistor of ~1k5 on a cathode-biased stage, there shouldn't be any drastic sonic differences and there definitely shouldn't be any performance problems with swapping the two types of tubes around in that circuit. The pin-out is the same, the heater current draw and heater voltage is the same, and even the VAmax is the same. (Granted the 12AT7 is rated as a higher dissipation tube, but all that means is that you should be able to run a 12AT7 hotter than you can run a 12AX7)

Having said that, some modern production tubes are coming out with appalling heater-cathode insulation, which can cause all sorts of problems (least of which is unwanted noise). Could be you had 2 bad tubes in a row?

What about other 12AX7s, 12AY7s, 5751s, 12BZ7s? You could try all those types in that circuit and you shouldn't have any problems. (But neither should you with 12AT7s)